Cuckoo Hashing for Undergraduates
Rasmus Pagh
IT University of Copenhagen
March 27, 2006
Abstract
This lecture note presents and analyses two simple hashing algorithms:
“Hashing with Chaining”, and “Cuckoo Hashing”. The analysis uses only
very basic (and intuitively understandable) concepts of probability theory,
and is meant to be accessible even for undergraduates taking their first
algorithms course.
1
Introduction
A dictionary is a data structure for storing a set of items (e.g., integers or
strings), that supports three basic operations: Lookup(x) which returns true
if x is in the current set, and false otherwise; Insert(x) which adds the item
x to the current set if not already present; Delete(x) which removes x from
the current set if present. A common variant is to associate some information
with each item, which is returned in connection with lookups. It is a simple
exercise to augment the algorithms presented here with this capability, but we
will describe just the basic variant.
One simple, but not very efficient implementation of a dictionary is a linked
list. In this implementation all operations take linear time in the worst case
(and even in the average case), assuming that insertions first check whether the
item is in the current list. A more scalable implementation of a dictionary is a
balanced search tree. In this lecture note we present two even more efficient data
structures based on hashing. The performance bounds of the algorithms will
not hold in the worst case, but be true in the “expected case”. This is because
the algorithms are not deterministic, but make random choices. The expected
performance is the average one over all random choices. More specifically, the
behavior of the algorithms will be guided by one, respectively two, hash functions
that take items as input, and return “random” values in some set {1, . . . , r}. To
simplify the exposition we make the following assumptions:
• All items to be stored have the same size, and we can compare two items
in constant time.
1
• We have access to hash functions h1 and h2 such that any function value
hi (x) is equal to a particular value in {1, . . . , r} with probability1 1/r,
and the function values are independent of each other (one function value
says nothing about other function values). Hash function values can be
computed in constant time.
• There is a fixed upper bound n on the number of items in the set.
The space usage of the algorithms will be bounded in terms of n. Specifically,
the space usage will be the same as for storing O(n) items, i.e., within a constant
factor of the space for storing the items with no space wasted.
We will see that the logarithmic time bounds of balanced search trees can be
improved to constant expected time per operation. In other words, the time for
an operation does not grow even if we fill the dictionary with sets of astronomical
size!
2
Hashing with Chaining
The main idea in hashing based dictionaries
is to let the hash functions decide where to
store each item. An item x will be stored at
“position h1 (x)” in an array of size r ≥ n.
Note that for this to work, it is crucial that
h1 is really a function, i.e., that h1 (x) is a
fixed value. However, it is highly likely that
there will be collisions, i.e., different items x
and y for which h1 (x) = h1 (y). So for each
value a ∈ {1, . . . , r} there is some set Sa
of items having this particular value of h1 .
An obvious idea is to make a pointer from
position a in the array to a data structure
holding the set Sa . Such a data structure
is called a bucket. Surprisingly, it suffices
to use a very simple (and inefficient) data
structure, namely a linked list (also known
as a “chained list”). An instance of the resulting data structure is shown in Figure 1.
Intuitively, the reason that linked lists work
well is that the sets are very small on average. In the following we will make a precise
analysis.
A
H
P
J
B
M
Q
D
Figure 1.
W
Hashing with chaining.
The items J, B and M have the same
value of h1 and have been placed in
the same “bucket”, a linked list of
length 3, starting at position h1 (J)
in the array.
We start out with two observations that will go into the analysis:
1 Note that this is only possible if hash functions are somehow chosen in a random fashion.
However, in this lecture note we will not describe how to do this.
2
1. For any two distinct items x and y, the probability that x hashes to the
bucket of y is O(1/r). This follows from our assumptions on h1 .
2. The time for an operation on an item x is bounded by some constant times
the number of items in the bucket of x.
The whole analysis will rest only on these observations. This will allow us to
reuse the analysis for Cuckoo Hashing, where the same observations hold for a
suitable definition of “bucket”.
Let us analyze an operation on item x. By observation 2, we can bound
the time by bounding the expected size of the bucket of x. For any operation,
it might be the case that x is stored in the data structure when the operation
begins, but this can cost only constant time extra, compared to the case in
which x is not in the list. Thus, we may assume that the bucket of x contains
only items different from x. Let S be the set of items that were present at
the beginning of the operation. For any y ∈ S, observation 1 says that the
probability that the operation spends time on y is O(1/r). Thus, the expected
(or “average”) time consumption attributed to y is O(1/r). To get the total
expected time, we must sum up the expected time usage for all elements in S.
This is |S| · O(1/r), which is O(1) since we chose r such that r ≥ n ≥ |S|. In
conclusion, the expected time for any operation is constant.
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Cuckoo Hashing
Suppose that we want to be able to look up items in constant time in the worst
case, rather than just in expectation. What could be done to achieve this? One
approach is to maintain a hash function that has no collisions for elements in
the set. This is called a perfect hash function, and allows us to insert the items
directly into the array, without having to use a linked list. Though this approach
can be made to work, it is quite complicated, in particular when dealing with
insertions of items. Here we will consider a simpler way, first described in
(R. Pagh and F. Rodler, Cuckoo Hashing, Proceedings of European Symposium
on Algorithms, 2001). In these notes the algorithm is slightly modified to enable
a simplified analysis.
Instead of requiring that x should be stored at position h1 (x), we give two
alternatives: Position h1 (x) and position h2 (x). We allow at most one element
to be stored at any position, i.e., there is no need for a data structure holding
colliding items. This will allow us to look up an item by looking at just two
positions in the array. When inserting a new element x it may of course still
happen that there is no space, since both position h1 (x) and position h2 (x) can
be occupied. This is resolved by imitating the nesting habits of the European
cuckoo: Throw out the current occupant y of position h1 (x) to make room!
This of course leaves the problem of placing y in the array. If the alternative
position for y is vacant, this is no problem. Otherwise y, being a victim of
the ruthlessness of x, repeats the behavior of x and throws out the occupant.
This is continued until the procedure finds a vacant position or has taken too
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long. In the latter case, new hash functions are chosen, and the whole data
structure is rebuilt (“rehashed”). The pseudocode for the insertion procedure
is shown in Figure 2. The variable “pos” keeps track of the position in which
we are currently trying to insert an item. The notation a ↔ b means that the
contents of a and b are swapped. Note that the procedure does not start out
with examining whether any of positions h1 (x) and h2 (x) are vacant, but simply
inserts x in position h1 (x).
A
C
B
H
P
W
procedure insert(x)
if T [h1 (x)] = x or T [h2 (x)] = x then return;
pos ← h1 (x);
loop n times {
if T [pos] = NULL then { T [pos] ← x; return};
x ↔ T [pos];
if pos= h1 (x) then pos← h2 (x) else pos← h1 (x);}
rehash(); insert(x)
end
Figure 2. Cuckoo hashing insertion procedure and illustration.
The arrows show the alternative position of each item in the dictionary. A new item would be inserted in the position of A by
moving A to its alternative position, currently occupied by B,
and moving B to its alternative position which is currently vacant. Insertion of a new item in the position of H would not
succeed: Since H is part of a cycle (together with W), the new
item would get kicked out again.
3.1
Examples and discussion
Figure 2 shows an example of the cuckoo graph, which is a graph that has an
edge for each item in the dictionary, connecting the two alternative positions of
the item. In the figure we have put an arrow on each edge to indicate which
element can move along the edge to its alternative position. We can note that
the picture has some similarities with Hashing with Chaining. For example, if
we insert an item in the position of A, the insertion procedure traverses the
path, or chain, involving the positions of A and B. Indeed, it is a simple exercise
to see that when inserting an item x, the insertion procedure will only visit
positions to which there is a path in the cuckoo graph from either position
h1 (x) or h2 (x). Let us call this the bucket of x. The bucket may have a more
complicated structure than in Hashing with Chaining, since the cuckoo graph
can have cycles. The graph in Figure 2, for example, has a cycle involving items
H and W. If we insert Z where h1 (Z) is the position of W, and h2 (Z) the position
of A, the traversal of the bucket will move items W, H, Z, A, and B, in this
order. In some cases there is no room for the new element in the bucket, e.g.,
if the possible positions are those occupied by H and W in Figure 2. In this
case the insertion procedure will loop n times before it gives up and the data
structure is rebuilt (“rehashed”) with new, hopefully better, hash functions.
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Observe that the insertion procedure can only loop n times if there is a cycle in
the cuckoo graph. In particular, every insertion will succeed as long as there is
no cycle. Also, the time spent will be bounded by a constant times the size of
the bucket, i.e., observation 2 from the analysis of Chained Hashing holds.
3.2
Analysis
By the above discussion, we can show that the expected insertion time (in
absence of a rehash) is constant by arguing that observation 1 holds as well.
In the following, we consider the undirected cuckoo graph, where there is no
orientation on the edges. Observe that y can be in the bucket of x only if there
is a path between one of the possible positions of x and the position of y in the
undirected cuckoo graph. This motivates the following lemma, which bounds
the probability of such paths:
Lemma 1 For any positions i and j, and any c > 1, if r ≥ 2cn then the
probability that in the undirected cuckoo graph there exists a path from i to j of
length ` ≥ 1, which is a shortest path from i to j, is at most c−` /r.
Interpretation. Before showing the lemma, let us give an interpretation in
words of its meaning. The constant c can be any number greater than 1, e.g. you
may think of the case c = 2. The lemma says that if the number r of nodes is
sufficiently large compared to the number n of edges, there is a low probability
that any two nodes i and j are connected by a path. The probability that they
are connected by a path of constant length is O(1/r), and the probability that
a path of length ` exists (but no shorter path) is exponentially decreasing in `.
Proof:
We proceed by induction on `. For the base case ` = 1 observe
that there is a set S of at most n items that may have i and j as their possible
positions. For each item, the probability that this is true is at most 2/r2 , since
at most 2 out of r2 possible choices P
of positions give the alternatives i and j.
Thus, the probability is bounded by x∈S 2/r2 ≤ 2n/r2 = c−1 /r.
In the inductive step we must bound the probability that there exists a path
of length ` > 1, but no path of length less than `. This is the case only if, for
some position k:
• There is a shortest path of length `−1 from i to k that does not go through
j, and
• There is an edge from k to j.
By the induction hypothesis, the probability that the first condition is true is
bounded by c`−1 /r. Given that the first condition is true, what is the probability
that the second one holds as well, i.e., that there exists an item that has k and
j as its possible positions? As before, there is a set S of less than n items that
could have j as one of their alternative positions, and for each one of them the
probability is at most 2/r2 . Thus, the probability is bounded by c−1 /r, and the
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probability that both conditions hold for a particular choice of k is less than
c` /r2 . Summing over the r possibilities for k, we get that the probability of a
path of length ` is at most c−` /r.
2
To show that observation 1 holds, we observe that if x and y are in the
same bucket, then there is a path of some length ` between {h1 (x), h2 (x)} and
{hP
1 (y), h2 (y)}. By the above lemma, this happens with probability at most
∞
4
/r = O(1/r), as desired.
4 `=1 c−` /r = c−1
Rehashing. We will end with a discussion of the cost of rehashing. We consider a sequence of operations involving n insertions, where is a small constant,
e.g. = 0.1. Let S 0 denote the set of items that exist in the dictionary at some
time during these insertions. Clearly, a cycle can exist at some point during
these insertions only if the undirected cuckoo graph corresponding to the items
in S 0 contain a cycle. A cycle, of course, is a path from a position to itself, so
Lemma 1Psays that any particular position is involved in a cycle with probability
∞
at most `=1 c−` /r, if we take r ≥ 2c(1 + )n. Thus, the probability
that there
P∞
1
is a cycle can be bounded by summing over all r positions: r· `=1 c−` /r = c−1
.
For c = 3, for example, the probability is at most 1/2 that a cycle occurs during
the n insertions. The probability that there will be two rehashes, and thus two
independent cycles is
and so on. In conclusion, the expected number of
P1/4,
∞
rehashes is at most i=1 2−i = 1. If the time for one rehashing is O(n), then
the expected time for all rehashings is O(n), which is O(1/) per insertion. This
means that the expected amortized cost of rehashing is constant. That a single
rehashing takes expected O(n) time follows from the fact that the probability
of a cycle in a given attempt is at most 1/2.
3.3
Final remarks
The analysis of rehashing above is not exact. In fact, the expected amortized
cost of rehashing is only O(1/n) per insertion, for any c > 1.
To get an algorithm that adapts to the size of the set stored, a technique
called global rebuilding can be used. Whenever the size of the set becomes too
small compared to the size of the hash table, a new, smaller hash table is created.
Conversely, if the hash table fills up to its capacity, a new, larger hash table
can be created. To make this work efficiently, the size of the hash table should
always be increased or decreased with a constant factor (larger than 1), e.g.
doubled or halved. Then the expected amortized cost of rebuilding is constant.
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